Numbers of Sides
'Many cheerful facts about the square of the hypotenuse,' and beyond.
Feb 25, 2008, Vol. 13, No. 23 • By DAVID GUASPARI
This seems to have received powerful support from the discovery, attributed to Pythagoras, that basic musical intervals are "rational." Stop a tensed violin string somewhere in the middle and consider the difference between the pitches produced by plucking its two parts. If the ratio between the lengths of those parts is two to one, the pitch difference is an octave; if two to three, it's a perfect fifth; and so on.
Note that the ratio of two lengths is the same as the ratio of two counting numbers precisely when there is a unit of which both lengths are exact multiples. An irrefutable proof that the sides and diagonal of a square are, in this sense, "irrational"--and that irrationality is an essential feature of the mathematical world--can only have been a metaphysical blow.
The first section of Maor's book stretches from the Babylonians to Archimedes, the greatest ancient mathematician, and one of the greatest ever. Insofar as it has a single theme, this section asks which societies knew of the Pythagorean Theorem, and in what form and in what way they knew it. The next thousand or so years get brief treatment as an interlude, an era of "translators and commentators"--illustrated by episodes from Chinese, Hindu, and Arabic, as well as Western, mathematics.
The final section begins in the mid-16th century with François Viète--often regarded as the first modern mathematician--and tells two related stories. One is the introduction of infinite methods and infinities into mathematics--controversial but successful innovations that had to wait 300 years for a rigorous basis. The other develops the "non-Euclidean" geometry that plays a central role in modern physics as the mathematical setting for Einstein's theory of general relativity.
The Pythagorean Theorem holds for figures drawn on a flat surface--that is, for the objects of Euclidean geometry. In other settings--figures drawn on the surface of a sphere, for example--it fails. The Theorem is a characteristic of flatness, hence its ubiquity: The calculations of trigonometry, of the lengths of lines (straight or curved), etc., are all intimately tied to it. The converse insight, that the geometry of a surface can be captured by describing the ways in which it deviates from the Pythagorean Theorem, makes it possible to represent and reason about unvisualizable geometries such as the "curved space-time" of Einstein's theory.
Maor ventilates these stories with frequent digressions. One (rather dull) chapter called "The Pythagorean Theorem in Art, Poetry, and Prose" provides a laundry list. There is the patter song from The Pirates of Penzance in which the modern major general boasts of his acquaintance "with many cheerful facts about the square of the hypotenuse." There are encomia to Pythagoras from Johannes Kepler and (descending from the sublime) Jacob Bronowski. There is the famous story of Thomas Hobbes's exclamation, on first seeing the Pythagorean Theorem: "By God, this is impossible!"
Another chapter presents excerpts from the curious life work of Elisha Scott Loomis, who undertook to gather all known proofs of the Pythagorean Theorem--of which he found 371, including one by President James Garfield (before his election). Also included are brain teasers, mathematical curiosities, and a short essay on the possibility of composing a message that would be understood by intelligent life in distant galaxies.
Bits of potted history serve as glue. Not all of this is reliable, as when Plato's contribution to geometry is described as "his recognition of its importance to learning in general, to logical thinking, and, ultimately, to a healthy democracy." It is not likely that Plato was fond of diseased democracy, but safe to say that promoting democracy was not one of his concerns. Plato's interest in geometry was metaphysical: The relation between ideal geometric figures (grasped by reason) and the imperfect copies that we draw or otherwise encounter through our senses prefigures the relation between a Platonic form, such as Goodness, and its imperfect realizations in the world of ordinary experience.
It is hard to predict who would be charmed by The Pythagorean Theorem, but all will recognize the author's enthusiasm for his subject and his respect for the reader: Three cheers for including those proofs!
David Guaspari is a writer in Ithaca, New York.